Let’s consider the first order linear differential equation \(y^{\,\prime }=2x\left (e^{-x^2}-2y\right )\).

(a)
Here is an (approximate) direction field for this differential equation on
\([-2,2]\times [-2,2]\). Let’s draw (part of) the solution curve to this differential equation that passes through the point \((0,1)\). Let’s use our drawing to estimate the value of this solution when \(x=1\).
A direction field.
Figure 1: A direction field
(b)
Let’s write this equation in standard form.
(c)
Let’s determine an integrating factor.
(d)
Now let’s solve this differential equation for its general solution, and determine the solution that passes through \((0,1)\).
(e)
How good was our estimate?